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Two samples of an AM signal with carrier frequency $\omega_c$.

\begin{align*}
\begin{cases}
x[0]=f(t_{0})\sin(\omega_c t_{0}+\alpha)\\
x[1]=f(t_{1})\sin(\omega_c t_{1}+\alpha)
\end{cases}
\end{align*}

If the samples were taken $\Delta t$ seconds apart, then the sampling frequency
$f_s$ is $1/\Delta t$.

\begin{align*}
\begin{cases}
x[0]=f(t_{0})\sin(\omega_c t_{0}+\alpha)\\
x[1]=f(t_{0}+\Delta t)\sin(\omega_c (t_{0}+\Delta t)+\alpha)
\end{cases}
\end{align*}

If $\Delta t$ is quite small, $f(t_0) = f(t_0 + \Delta t) = A$: 

\begin{align*}
\begin{cases}
x[0]=A\sin(\omega_c t_{0}+\alpha)\\
x[1]=A\sin(\omega_c (t_{0}+\Delta t)+\alpha)
\end{cases}
\end{align*}

If $t_0 = 0$:

\begin{align*}
\begin{cases}
x[0]=A\sin(\alpha)\\
x[1]=A\sin(\omega_c \Delta t+\alpha)
\end{cases}
\end{align*}

We define $\phi$ as $\omega_c \Delta t = \omega_c / f_s = 2\pi f_c / f_s$.

\begin{align*}
\begin{cases}
x[0]=A\sin(\alpha)\\
x[1]=A\sin(\phi+\alpha)
\end{cases}
\end{align*}
\begin{align*}
\begin{cases}
\alpha=\arcsin\left(\frac{x[0]}{A}\right)\\
\alpha=\arcsin\left(\frac{x[1]}{A}\right)-\phi
\end{cases}
\end{align*}
\begin{align*}
\arcsin\left(\frac{x[0]}{A}\right)=\arcsin\left(\frac{x[1]}{A}\right)-\phi
\end{align*}
\begin{align*}
\frac{x[0]}{A}=\sin\left(\arcsin\left(\frac{x[1]}{A}\right)-\phi\right)
\end{align*}

\begin{align*}
\frac{x[0]}{A}=\frac{x[1]}{A}\cos\phi-\cos\left(\arcsin\left(\frac{x[1]}{A}\right)\right)\sin\phi
\end{align*}

\begin{align*}
\frac{x[0]}{A}=\frac{x[1]}{A}\cos\phi-\sqrt{1-\frac{x[1]^{2}}{A^{2}}}\sin\phi
\end{align*}

\begin{align*}
x[0]=x[1]\cos\phi-\sqrt{A^{2}-x[1]^{2}}\sin\phi
\end{align*}

\begin{align*}
A^{2}-x[1]^{2}=\left(\frac{x[1]\cos\phi-x[0]}{\sin\phi}\right)^{2}
\end{align*}

\begin{align*}
A^{2}-x[1]^{2}=\frac{x[1]^{2}\cos^{2}\phi+x[0]^{2}-2x[1]x[0]\cos\phi}{\sin^{2}\phi}
\end{align*}

\begin{align*}
A^{2}=\frac{x[1]^{2}\cos^{2}\phi+x[1]^{2}\sin^{2}\phi+x[0]^{2}-2x[1]x[0]\cos\phi}{\sin^{2}\phi}
\end{align*}

\begin{align*}
A=\frac{\sqrt{x[1]^{2}+x[0]^{2}-2x[1]x[0]\cos\phi}}{\sin\phi}
\end{align*}

Then:

\begin{align*}
y[i]=\frac{\sqrt{x[i]^{2}+x[i-1]^{2}-2x[i]x[i-1]\cos\phi}}{\sin\phi}
\end{align*}

Where $\phi = 2\pi f_c / f_s$ 

\end{document}
